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arXiv · 1910.04361

Tree automata and pigeonhole classes of matroids: II

Daryl Funk·Dillon Mayhew·Mike Newman
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Abstract

Let $\psi$ be a sentence in the counting monadic second-order logic of matroids and let $\mathbb{F}$ be a finite field. Hlin\v{e}n\'{y}'s Theorem says that we can test whether $\mathbb{F}$-representable matroids satisfy $\psi$ using an algorithm that is fixed-parameter tractable with respect to branch-width. In a previous paper we proved there is a similar fixed-parameter tractable algorithm that can test the members of any efficiently pigeonhole class. In this sequel we apply results from the first paper and thereby extend Hlin\v{e}n\'{y}'s Theorem to the classes of fundamental transversal matroids, lattice path matroids, bicircular matroids, and $H$-gain-graphic matroids, when $H$ is a finite group. As a consequence, we can obtain a new proof of Courcelle's Theorem.

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Daryl Funk, Dillon Mayhew, Mike Newman. 2019-10-10. Tree automata and pigeonhole classes of matroids: II. https://arxiv.org/abs/1910.04361

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